from the category of smooth projective varieties over kk given by mapping X↦XX \mapsto X and a morphism f:X→Yf : X \to Y to its graph, the image of its graph morphismΓf:X→X×Y\Gamma_f : X \to X \times Y.

Category of effective pure motives

The Karoubi envelope (pseudo-abelianisation) of Corr∼(k,A)Corr_\sim(k, A) is called the category of effective pure motives (with coefficients in AA and with respect to the equivalence relation ∼\sim), denoted Mot∼eff(k,A)Mot^eff_\sim(k, A).

Explicitly its objects are pairs (h(X),p)(h(X), p) with XX a smooth projective variety and p∈Corr(h(X),h(X))p \in Corr(h(X), h(X)) an idempotent, and morphisms from (h(X),p)(h(X), p) to (h(Y),q)(h(Y), q) are morphisms h(X)→h(Y)h(X) \to h(Y) in Corr∼Corr_\sim of the form q∘α∘pq \circ \alpha \circ p with α∈Corr∼(h(X),h(Y))\alpha \in Corr_{\sim}(h(X), h(Y)).